Using Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is [latex]20[/latex] years old and Alex is [latex]23[/latex], so Alex is [latex]3[/latex] years older than Greg. When Greg was [latex]12[/latex], Alex was [latex]15[/latex]. When Greg is [latex]35[/latex], Alex will be [latex]38[/latex]. No matter what Greg’s age is, Alex’s age will always be [latex]3[/latex] years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The [latex]3[/latex] years between them always stays the same, so the age difference is the constant.

In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age [latex]g[/latex]. Then we could use [latex]g+3[/latex] to represent Alex’s age. See the table below.

To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. Earlier, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.

In algebra, the cross symbol, [latex]\times [/latex], is no longer used to show multiplication because that symbol may cause confusion with the letter x, or variable [latex]x[/latex]. Does 3xy mean [latex]3\times y[/latex] (three times [latex]y[/latex] ) or [latex]3\cdot x\cdot y[/latex] (three times [latex]x\text{ times }y[/latex] )? To make it clear to everyone, we use a center dot • or parentheses ( ) to represent multiplication in algebra.

Translating from words to algebra or vice versa is an important skill. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help find the numbers.

• The sum of [latex]5[/latex] and [latex]3[/latex] means addition,  [latex]5[/latex] plus [latex]3[/latex], which we write as [latex]5+3[/latex].
• The difference of [latex]9[/latex] and [latex]2[/latex] means subtraction,  [latex]9[/latex] minus [latex]2[/latex], which we write as [latex]9 - 2[/latex].
• The product of [latex]4[/latex] and [latex]8[/latex] means multiplication,  [latex]4[/latex] times [latex]8[/latex], which we can write as [latex]4 \times 8 [/latex].
• The quotient of [latex]20[/latex] and [latex]5[/latex] means division, divide [latex]20[/latex] by [latex]5[/latex], which we can write as [latex]20\div 5[/latex].

When two quantities have the same value, we say they are equal and connect them with an equals sign.